Solve (5sqrt{14}-3sqrt{21})^2

$$( 5 \sqrt { 14 } - 3 \sqrt { 21 } ) ^ { 2 }$$

Evaluate

$$539-210\sqrt{6}\approx 24.607154016$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5\sqrt{14}-3\sqrt{21}\right)^{2}.

$$25\left(\sqrt{14}\right)^{2}-30\sqrt{14}\sqrt{21}+9\left(\sqrt{21}\right)^{2}$$

The square of \sqrt{14} is 14.

$$25\times 14-30\sqrt{14}\sqrt{21}+9\left(\sqrt{21}\right)^{2}$$

Multiply 25 and 14 to get 350.

$$350-30\sqrt{14}\sqrt{21}+9\left(\sqrt{21}\right)^{2}$$

To multiply \sqrt{14} and \sqrt{21}, multiply the numbers under the square root.

$$350-30\sqrt{294}+9\left(\sqrt{21}\right)^{2}$$

The square of \sqrt{21} is 21.

$$350-30\sqrt{294}+9\times 21$$

Multiply 9 and 21 to get 189.

$$350-30\sqrt{294}+189$$

Add 350 and 189 to get 539.

$$539-30\sqrt{294}$$

Factor 294=7^{2}\times 6. Rewrite the square root of the product \sqrt{7^{2}\times 6} as the product of square roots \sqrt{7^{2}}\sqrt{6}. Take the square root of 7^{2}.

$$539-30\times 7\sqrt{6}$$

Multiply -30 and 7 to get -210.

$$539-210\sqrt{6}$$

Expand

$$539-210\sqrt{6}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5\sqrt{14}-3\sqrt{21}\right)^{2}.

$$25\left(\sqrt{14}\right)^{2}-30\sqrt{14}\sqrt{21}+9\left(\sqrt{21}\right)^{2}$$

The square of \sqrt{14} is 14.

$$25\times 14-30\sqrt{14}\sqrt{21}+9\left(\sqrt{21}\right)^{2}$$

Multiply 25 and 14 to get 350.

$$350-30\sqrt{14}\sqrt{21}+9\left(\sqrt{21}\right)^{2}$$

To multiply \sqrt{14} and \sqrt{21}, multiply the numbers under the square root.

$$350-30\sqrt{294}+9\left(\sqrt{21}\right)^{2}$$

The square of \sqrt{21} is 21.

$$350-30\sqrt{294}+9\times 21$$

Multiply 9 and 21 to get 189.

$$350-30\sqrt{294}+189$$

Add 350 and 189 to get 539.

$$539-30\sqrt{294}$$

Factor 294=7^{2}\times 6. Rewrite the square root of the product \sqrt{7^{2}\times 6} as the product of square roots \sqrt{7^{2}}\sqrt{6}. Take the square root of 7^{2}.

$$539-30\times 7\sqrt{6}$$

Multiply -30 and 7 to get -210.

$$539-210\sqrt{6}$$